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The eccentricity of the hyperbola with l...

The eccentricity of the hyperbola with latursrectum `12` and semi-conjugate axis is `2sqrt(3)`, is

A

`1`

B

`2`

C

`3`

D

`4`

Text Solution

Verified by Experts

Let the length of each side of the equilateral triangle `OPQ` be `l` units. Then, the coordinates of `P` are `((sqrt(3l))/(2),(l)/(2))`.
Point `P((sqrt(3l))/(2),(l)/(2))` lies on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`

`:. (3l^(2))/(4a^(2))-(l^(2))/(4b^(2))=1`
`implies(3b^(2)-a^(2))l^(2)=4a^(2)b^(2)`
`implies(3e^(2)-4)l^(2)=4a^(2)(e^(2)-1)impliesl=2asqrt((e^(2)-1)/(3e^(2)-4))`
Since `l` is real and `e gt 1`.
`:.3e^(2)-4 gt 0impliese^(2) gt (4)/(3)impliese gt (2)/(sqrt(3))`
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